We discuss some of the key themes in dynamical systems and especially in holomorphic dynamics. One key theme will be Newton’s famous root finding method that is a prominent holomorphic map that “wants to be iterated” and thus naturally forms a dynamical system. These dynamical systems that can be investigated successfully from the point of view of a celebrated theory developed by William Thurston that treats the geometry of 3-manifolds, automorphisms of surfaces, and holomorphic dynamical systems in a similar way. We also discuss Newton’s method in practice as a root finder: while it traditionally had a reputation as “difficult to predict” because of its chaotic nature, we show that — unlike many root finding methods — it does have a successful theory, and it works extremely well in practice. Recently young students in our team found all roots for complex polynomials of degrees greater than 100 million.